An Extension of Geiringer's Theorem for a Wide Class of Evolutionary Search Algorithms
该文证明了一个通用定理：在温和条件下，无选择时进化算法种群马尔可夫链的平稳分布唯一且均匀，这推广了经典盖林格定理，并为更广的进化算法提供了类似结论的推导方法。
The frequency with which various elements of the search space of a given evolutionary algorithm are sampled is affected by the family of recombination (reproduction) operators. The original Geiringer theorem tells us the limiting frequency of occurrence of a given individual under repeated application of crossover alone for the classical genetic algorithm. Recently, Geiringer's theorem has been generalized to include the case of linear GP with homologous crossover (which can also be thought of as a variable length GA). In the current paper we prove a general theorem which tells us that under rather mild conditions on a given evolutionary algorithm, call it A, the stationary distribution of a certain Markov chain of populations in the absence of selection is unique and uniform. This theorem not only implies the already existing versions of Geiringer's theorem, but also provides a recipe of how to obtain similar facts for a rather wide class of evolutionary algorithms. The techniques which are used to prove this theorem involve a classical fact about random walks on a group and may allow us to compute and/or estimate the eigenvalues of the corresponding Markov transition matrix which is directly related to the rate of convergence towards the unique limiting distribution.